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Properties

Label 20.12.1712060409...7369.1
Degree $20$
Signature $[12, 4]$
Discriminant $11^{16}\cdot 43^{2}\cdot 67^{4}$
Root discriminant $23.00$
Ramified primes $11, 43, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 10, -5, -38, -46, 41, 484, -201, -1021, 563, 671, -474, -14, 47, -45, 45, -25, 12, 1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + x^18 + 12*x^17 - 25*x^16 + 45*x^15 - 45*x^14 + 47*x^13 - 14*x^12 - 474*x^11 + 671*x^10 + 563*x^9 - 1021*x^8 - 201*x^7 + 484*x^6 + 41*x^5 - 46*x^4 - 38*x^3 - 5*x^2 + 10*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + x^18 + 12*x^17 - 25*x^16 + 45*x^15 - 45*x^14 + 47*x^13 - 14*x^12 - 474*x^11 + 671*x^10 + 563*x^9 - 1021*x^8 - 201*x^7 + 484*x^6 + 41*x^5 - 46*x^4 - 38*x^3 - 5*x^2 + 10*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + x^{18} + 12 x^{17} - 25 x^{16} + 45 x^{15} - 45 x^{14} + 47 x^{13} - 14 x^{12} - 474 x^{11} + 671 x^{10} + 563 x^{9} - 1021 x^{8} - 201 x^{7} + 484 x^{6} + 41 x^{5} - 46 x^{4} - 38 x^{3} - 5 x^{2} + 10 x - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1712060409270190644695047369=11^{16}\cdot 43^{2}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.00$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $11, 43, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{12} a^{15} - \frac{1}{4} a^{14} - \frac{1}{6} a^{13} - \frac{1}{12} a^{12} + \frac{1}{6} a^{11} - \frac{1}{12} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{12}$, $\frac{1}{12} a^{16} + \frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{12} a^{5} + \frac{1}{3} a^{4} + \frac{1}{4} a^{3} + \frac{1}{6} a - \frac{1}{4}$, $\frac{1}{12} a^{17} + \frac{1}{6} a^{14} + \frac{1}{12} a^{13} - \frac{1}{4} a^{11} - \frac{1}{12} a^{10} - \frac{1}{3} a^{9} + \frac{5}{12} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{5}{12} a^{5} + \frac{1}{12} a^{4} + \frac{1}{6} a^{3} - \frac{1}{12} a^{2} - \frac{1}{2} a + \frac{1}{12}$, $\frac{1}{12} a^{18} + \frac{1}{12} a^{14} - \frac{1}{6} a^{13} - \frac{1}{12} a^{12} + \frac{1}{12} a^{11} - \frac{1}{6} a^{10} + \frac{1}{12} a^{9} - \frac{1}{2} a^{7} + \frac{1}{12} a^{6} - \frac{5}{12} a^{5} - \frac{1}{6} a^{4} - \frac{1}{4} a^{3} + \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{8791834998151668} a^{19} + \frac{120969392786969}{2930611666050556} a^{18} - \frac{60217729937281}{2930611666050556} a^{17} - \frac{69050287141561}{2930611666050556} a^{16} + \frac{39233251127353}{1465305833025278} a^{15} + \frac{1513218762831415}{8791834998151668} a^{14} - \frac{606364934766089}{8791834998151668} a^{13} - \frac{798761674404449}{4395917499075834} a^{12} + \frac{1947494468234561}{8791834998151668} a^{11} - \frac{399806377227695}{4395917499075834} a^{10} - \frac{1066868263683851}{8791834998151668} a^{9} - \frac{3526297183522345}{8791834998151668} a^{8} - \frac{223471776920127}{732652916512639} a^{7} + \frac{3216269181201307}{8791834998151668} a^{6} + \frac{814833678743920}{2197958749537917} a^{5} + \frac{818560460799106}{2197958749537917} a^{4} - \frac{39724339558091}{4395917499075834} a^{3} - \frac{3400468178691251}{8791834998151668} a^{2} - \frac{537000660077683}{2197958749537917} a - \frac{2020296635212259}{8791834998151668}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2236032.59751 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.617567936161.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
43Data not computed
67Data not computed